Download Slutsky Matrix Norms: The Size, Classification, and Comparative

Slutsky Matrix Norms: The Size, Classification, and
Comparative Statics of Bounded Rationality∗
Victor H. Aguiar†and Roberto Serrano‡
This version: August 2015
Abstract
Given any observed demand behavior by means of a demand function, we quantify by
how much it departs from rationality. The measure of the gap is the smallest Frobenius
norm of the correcting matrix function that would yield a Slutsky matrix with its standard rationality properties (symmetry, singularity, and negative semidefiniteness). A useful
classification of departures from rationality is suggested as a result. Errors in comparative
statics predictions from assuming rationality are decomposed as the sum of a behavioral
error (due to the agent) and a mis-specification error (due to the modeller). Comparative
statics for the boundedly rational consumer is expressed as path dependence of wealth compensations in price-equivalent paths, or as violations of the compensated law of demand.
Illustrations are provided using several bounded rationality models.
JEL classification numbers: C60, D10.
Keywords: consumer theory; rationality; Slutsky matrix function; bounded rationality;
comparative statics.
1
Introduction
The rational consumer model has been at the heart of most theoretical and applied work in
economics. In the standard theory of the consumer, this model has a unique prediction in the
form of a symmetric, singular, and negative semidefinite Slutsky matrix. In fact, any demand
system that has a Slutsky matrix with these properties can be viewed as being generated as the
result of a process of maximization of some rational preference relation. Nevertheless, empirical
evidence often derives demand systems that conflict with the rationality paradigm. In such
cases, those hypotheses (e.g., symmetry of the Slutsky matrix) are rejected. These important
findings have given rise to a growing literature of behavioral models that attempt to better fit
the data.
At this juncture three related questions can be posed in this setting:
∗ This
paper subsumes Aguiar and Serrano (2014). We are especially indebted to Xavier Gabaix, Michael
Jerison, and Joel Sobel for suggesting many specific improvements to that earlier version of the paper. We
also thank Bob Anderson, Francis Bloch, Mark Dean, Federico Echenique, Drew Fudenberg, Peter Hammond,
Susanne Schennach, Larry Selden, Jesse Shapiro and the participants at several conferences and seminars for
helpful comments and encouragement.
† Department of Economics, Brown University, victor [email protected]
‡ Department of Economics, Brown University, roberto [email protected]
1
ˆ (i) How can one measure the distance of an observed demand behavior –demand function–
from rationality?
ˆ (ii) How can one compare and classify two behavioral models as departures from a closest
rational approximation?
ˆ (iii) Given an observed demand function, what is the best rational approximation model?
The aim of this paper is to provide a tool to answer these three questions in the form of a Slutsky
matrix function norm, which allows to measure departures from rationality in either observed
Slutsky matrices or demand functions. The answer, provided for the class of demand functions
that are continuously differentiable, sheds light on the size and type of bounded rationality that
each observed behavior exhibits.
Our primitive is an observed demand function. To measure the gap between that demand
function and the set of rational behaviors, one can use the “least” distance and try to identify
the closest rational demand function. This approach presents serious difficulties, though. Leaving aside compactness issues, which can be addressed under some regularity assumptions, the
solution would require solving a challenging system of partial differential equations. Lacking
symmetry of this system, an exact solution may not exist, and one needs to resort to approximation or computational techniques, but those are still quite demanding.
We take an alternative approach, based on the calculation of the Slutsky matrix function of
the observed demand. We pose a matrix nearness problem in a convex optimization framework,
which permits both a better computational implementability and the derivation of extremum
solutions. Indeed, we attempt to find the smallest correcting additive perturbation to the observed Slutsky matrix function that will yield a matrix function with all the rational properties
(symmetry, singularity with the price vector on its null space, and negative semidefiniteness).
We use the Frobenius norm to measure the size of this additive factor, interpreting it as the
“size” of the observed departure from rationality.
We provide a closed-form solution to the matrix nearness problem just described. Interestingly, the solution can be decomposed into three separate terms, whose intuition we provide
next. Given an observed Slutsky matrix function,
ˆ (a) the norm of its anti-symmetric or skew symmetric part measures the “size” of the
violation of symmetry;
ˆ (b) the norm of the smallest additive matrix that will make the symmetric part of the
Slutsky matrix singular measures the “size” of the violation of singularity; and
ˆ (c) the norm of the positive semidefinite part of the resulting corrected matrix measures
the “size” of the violation of negative semidefiniteness.
Our main result shows that the “size” of bounded rationality, measured by the Slutsky matrix
squared norm, is simply the sum of the squares of these three effects. In particular, following any
observed behavior, we can classify the instances of bounded rationality as violations of the Ville
axiom of revealed preference –VARP– if only symmetry fails, violations of homogeneity of degree
zero or other money illusion phenomena if only singularity fails, violations of the weak axiom of
revealed preference –WARP– by a symmetric consumer if only negative semidefiniteness fails,
2
or combinations thereof in more complex failures, by adding up the nonzero components of the
norm.
The size of bounded rationality provided by the Slutsky norm depends on the units in which
the consumption goods are expressed. It is therefore desirable to provide unit-independent
measures, and we do so following the approach of modifying the Slutsky matrix by a weighting
matrix. For example, one can translate the first norm into dollars, hence providing a monetary
measure, or one can instead use a budget shares version that is unit-free.
The Slutsky matrix function is the key object in comparative statics analysis in consumer
theory. It encodes all the information of local demand changes with respect to small Slutsky
compensated price changes. We obtain comparative statics results for a boundedly rational
consumer. Importantly, one can decompose the error in comparative statics from assuming a
given form of rationality as the sum of two independent terms. The first is the behavioral error
(measured by our Slutsky norm and its decomposition), and the second is a mis-specification error
given the assumed parameterized rationality model. Further, unlike her rational counterpart, a
boundedly rational consumer may exhibit path dependence of wealth compensations in priceequivalent paths or may violate the compensated law of demand, and we relate such phenomena
to the different nonzero terms in our decomposition of the norm.
The rest of this paper is organized as follows. Section 2 presents the model. Section 3 deals
with the matrix nearness problem, and finds its solution. Section 4 emphasizes its additive
decomposition, and provides interpretations of the matrix nearness problem in terms of the
axioms behind revealed preference. Section 5 presents weighted Slutsky norms. Section 6 goes
over comparative statics. Section 7 presents several examples and applications of the result,
including hyperbolic discounting and the sparse consumer model. Section 8 is a brief review of
the literature, and Section 9 concludes. Some of the proofs of the more technical results are
collected in an appendix.
2
The Model
Consider a demand function x : Z 7→ X, where Z ≡ P × W is the compact space of price-wealth
L
pairs (p, w), P ⊆ RL
++ , W ⊆ R++ , and X ≡ R is the consumption set. This demand system is
a generic function that maps price and wealth to consumption bundles.
Assume that x(p, w) is continuously differentiable and satisfies Walras’ law: p0 x(p, w) = w for
all (p, w) ∈ Z. Let X ⊂ C 1 denote a set of functions that satisfy these characteristics. To be more
explicit noting the dependence on the domain, define also X (Z) ⊂ C 1 (Z), with C 1 (Z) denoting
the complete metric space of vector-valued functions f : Z 7→ RL , continuously differentiable,
1
uniformly bounded with compact domain Z ⊂ RL+1
++ , equipped with a norm.
Our attempt in this paper is to shed light on the size and types of bounded rationality.
To that end, let R ⊂ X be the set of rational demand functions (i.e., xr ∈ R is the solution
to maximizing a complete, locally nonsatiated and transitive preference over a linear budget
constraint). We define analogously R(Z) ⊂ X (Z) to be the set of allowable rational demand
functions.
1 For instance, the norm ||f ||
C = max({||fl ||C,1 }l=1,...,L ), with ||fl ||C,1 = max(||fl ||∞,1 ) where f (z) =
[f1 (z) · · · fL (z)]0 ∈ RL ). We use also the notation for the norm || · ||∞,m = supz∈Z |g(z)| for g : Z 7→ Rm , for
finite m ≥ 1 and | · | the absolute value. By the set C 1 (Z) we mean the set of functions that have C 1 extensions
to an open set containing the compact domain Z.
3
Definition 1. Define the distance of x ∈ X to the set of rational demands R by the “least”
distance from an element to a set: d(x, R) = inf {dX (x, xr )|xr ∈ R}.
We shall refer to this problem of trying to find the closest rational demand to a given demand
as the “behavioral nearness” problem (Varian, 1990). Observe that the behavioral nearness
problem at this level of generality presents several difficulties. First, the constraint set R(Z),
i.e., the set of rational demand functions, is not convex. In addition, the Lagrangian depends not
only upon xr but also on its partial derivatives. The typical curse of dimensionality of calculus
of variations applies here with full force, in the case of a large number of commodities. Indeed,
the Euler-Lagrange equations in this case do not offer much information about the problem and
give rise to a large partial differential equations system. Finally, to calculate analytically the
solution to this program is computationally challenging.2
Thus, instead of solving directly the “behavioral nearness” problem, our approach will rely on
matrix spaces. Our next goal is to talk about Slutsky norms. Let M(Z) be the complete metric
space of matrix-valued functions, F : Z 7→ RL × RL , equipped with the inner product hF, Gi =
´
´
T r(F (z)0 G(z))dz. This vector space has a Frobenius norm ||F ||2 = z∈Z T r(F (z)0 F (z))dz.3
z∈Z
Let us define the Slutsky substitution matrix function.
Definition 2. Let Z ⊂ P × W be given, and denote by z = (p, w) an arbitrary price-wealth
pair in Z.4 Then the Slutsky matrix function S ∈ M(Z) is defined pointwise: S(z) = Dp x(z) +
Dw x(z)x(z)0 ∈ RL×L , with entry sl,k (p, w) =
∂xl (p,w)
∂pk
+
∂xl (p,w)
xk (p, w).
∂w
The Slutsky matrix function is well defined for all x ∈ C 1 (Z). Restricted to the set of rational
behaviors, the Slutsky matrix satisfies a number of regularity conditions. Specifically, when a
matrix function S ∈ M(Z) is symmetric, negative semidefinite (NSD), and singular with p in
its null space for all z ∈ Z, we shall say that the matrix satisfies property R, for short.
Definition 3. For any Slutsky matrix function S ∈ M(Z), let its Slutsky norm be defined as
follows: d(S) = min{||E|| : S − E ∈ M(Z) having property R}.
The use of the minimum operator is justified. Indeed, it will be proved that the set of Slutsky
matrix functions satisfying R is a closed and convex set. Then, under the metric induced by the
Frobenius norm, the minimum will be attained in M(Z). We shall refer to the minimization
problem implied in the Slutsky norm as the “matrix nearness” problem.
3
The Matrix Nearness Problem: Its Solution
In this section we provide the exact solution to the matrix nearness problem, which allows us
to quantify the distance from rationality by measuring the size of the violations of the Slutsky
matrix conditions.
3.1
Preliminaries on Matrices
We begin by reviewing some definitions.
2 We refer the reader to our expanded discussion in Aguiar and Serrano (2014), which includes the use of the
“almost implies near” approach.
3 While one could use a different norm, we justify our choice of the Frobenius norm in the sequel.
4 Since Z is closed, we use the definition of differential of Graves (1956) that is defined not only in the interior
but also on the accumulation points of Z.
4
It will be useful to denote the three regularity conditions of any Slutsky matrix function with
shorthands. We shall use σ for symmetry, π for singularity with p in its null space (p−singularity),
and ν for negative semidefiniteness.
Given any square matrix-valued function, S ∈ M(Z), let S sym = 21 [S + S 0 ] be its symmetric
part. With our new notation, if S is a Slutsky matrix function, S σ = S sym . Equivalently, S σ is
the projection of the function S on the closed subspace of symmetric matrix-valued functions,
using the inner product defined for M(Z).
Every square matrix function S ∈ M(Z) can be written as S(z) = S sym (z) + S skew (z)
for z ∈ Z, also written as S = S σ + E σ , where S σ = S sym is its symmetric part and E σ =
S skew = 21 [S − S 0 ] is its anti-symmetric or skew-symmetric part (the orthogonal complement of
the symmetric part).
Every symmetric matrix function S σ ∈ M(Z), in particular the symmetric part of a Slutsky
matrix function S, can be decomposed into the sum of a singular and not singular parts (with
prices in the null space): see Claim 3.5 With our notation, the matrix function part that is
singular with p in its null space will be denoted S σ,π , that is, S σ,π (z)p = 0. Then S σ = S σ,π +E π
where S σ,π = P S σ P and E π = S σ − P S σ P is its orthogonal complement with P = I −
pp0
p0 p
a
projection matrix.6
Any symmetric matrix-valued function S σ ∈ M(Z) and in particular any matrix function
that is the singular in prices part S σ,π ∈ M(Z) of a Slutsky matrix can be pointwise decomposed
into the sum of its positive semidefinite and negative semidefinite parts. Indeed, we can always
write S σ,π (z) = S σ,π (z)+ + S σ,π (z)− , with S σ,π (z)+ S σ,π (z)− = 0 for all z ∈ Z. Moreover, for
any square matrix-valued function S ∈ M(Z), its projection on the cone of NSD matrix-valued
σ,π
functions under the Frobenius norm is S−
.
In general, a square matrix function may not admit diagonalization. However, we know
thanks to Kadison (1984) that every symmetric matrix-valued function in the set M(Z) is
diagonalizable. In particular, S σ,π can be diagonalized: S σ,π (z) = Q(z)Λ(z)Q(z)0 . Here, Λ(z) =
Diag[{λl (z)}l=1,...,L ], where Λ(z) ∈ M(Z), with λl : Z 7→ R a real-valued function with norm
|| · ||s (a norm in C 1 (R)), and λ1 ≤ λ2 ≤ . . . ≤ λL with the order derived from the metric induced
by the || · ||s norm.7 Let Q(z) = [q1 (z) . . . qL (z)], where Q ∈ M(Z) and its columns ql ∈ C 1 (Z)
are the eigenvector functions such that for l = 1, . . . , L: S σ,π (z)ql (z) = λl (z)ql (z) for z ∈ Z.
Any real-valued function can be written as λ(z) = λ(z)+ +λ(z)− , with λ(z)+ = max{λ(z), 0}
and λ(z)− = min{λ(z), 0}. This decomposition allows us to write S σ,π,ν (z) = S σ,π (z)− =
Q(z)Λ(z)− Q(z)0 for Λ(z)− = Diag[{λl (z)− }l=1,···L ] with λl (z)− the negative part function for
the λl (z) function. We can write also E ν (z) = S σ,π (z)+ = S σ,π (z) − S σ,π (z)− for z ∈ Z,
or S σ,π (z)+ = Q(z)Λ(z)+ Q(z)0 with Λ(z)+ defined analogously to Λ(z)− as the orthogonal
complement of S σ,π,ν .
3.2
The Result
We are ready to state the solution to the matrix nearness problem:
5 This means that we consider the space of symmetric matrix functions S σ that satisfy p0 S σ (z)p = 0. This is
a consequence of Walras’ law because p0 S(z) = 0, although not necessarily S(z)p = 0.
6 Because of Walras’ law when S is the Slutsky matrix of some demand function x ∈ X (Z), p0 S(z) = 0 and
S σ,π = S σ − E π and E π (z) = p10 p [S σ (z)pp0 + pp0 S σ (z)].
7 The order of the eigen-values is inessential to the results but it is convenient for the proofs. Observe that we
can have many diagonal decompositions all of which will work for our purposes.
5
Theorem 1. Given a Slutsky matrix S, it has the unique decomposition S = S r + E, where
S r = S σ,π,ν is the solution to the matrix nearness problem and E is the sum of the orthogonal
complements of the symmetric, singularity in p and NSD parts of S: E = E σ + E π + E ν .
Furthermore, the Slutsky norm of the solution to the matrix nearness problem can be decomposed
as follows:
||E||2 = ||E σ ||2 + ||E π ||2 + ||E ν ||2 .
We elaborate at length on the different components of this solution in the next section, right
after the proof of the theorem.
Proof. We first establish that the matrix nearness problem has a solution, and that it is unique.
This is done in Claim 1. Its proof is in the appendix.
Claim 1. The solution to the matrix nearness problem exists, and it is unique.
The rest of the proof of the theorem is done in two parts. Lemma 1 gives the solution imposing
only the singularity with p in its null space and symmetry restrictions. After that, Lemma 2
rewrites the problem slightly, and the solution is provided by adding the NSD restriction.
Lemma 1. The solution to minA ||S − A|| subject to A(z)p = 0, A(z) = A(z)0 for all z ∈ Z is
S σ,π .
Proof. The Lagrangian for the subproblem with symmetry and singularity restrictions is: L =
´
´
´
T r([S(z) − A(z)]0 [S(z) − A(z)])dz + z∈Z λ(z)0 A(z)pdz + z∈Z vec(U (z))0 vec[A(z) − A(z)0 ],
z∈Z
with λ ∈ RL and U ∈ RL × RL . Using that the singularity restriction term is scalar
(λ0 S σ,π (z)p ∈ R), as well as the identity T r(A0 B) = vec(A)0 vec(B) for all A, B ∈ M(Z),8
one can rewrite the Lagrangian as:
´
L = z∈Z [T r([S(z) − A(z)]0 [S(z) − A(z)]) + T r(A(z)pλ(z)0 ) + T r(U (z)0 [A(z) − A(z)0 ])]dz
Using the linearity of the trace, and the fact that this calculus of variations problem does
not depend on z or on the derivatives of the solution S σ,π , the pointwise first-order necessary
and sufficient conditions in this convex problem (Euler Lagrange Equation) is:
S(z) − A(z) + U (z) − U (z)0 + λ(z)p0 = 0.
Solve for A(z):
A(z) = S(z) + U (z) − U (z)0 + λ(z)p0 .
By the symmetry restriction,
2[U (z) − U (z)0 ] = S(z)0 − S(z) + pλ(z)0 − λ(z)p0 .
Replace back in the expression for A(z):
A(z) = S(z) + 12 [S(z)0 − S(z) + pλ(z)0 − λ(z)p0 ] + λ(z)p0 ,
We obtain the expression:
A(z) = S σ (z) + λ(z)p0 + 21 [pλ(z)0 − λ(z)p0 ].
Now we observe that we can write,
A(z) = S σ (z) + W σ (z)with W (z) = λ(z)p0 and W σ = 21 [pλ(z)0 + λ(z)p0 ], the symmetric part
of W (z).
By the restriction of singularity in prices A(z)p = 0 which implies W σ (z)p = −S σ (z)p.
8 The vec(A) symbol stands for the vectorization of a matrix A of dimension L × L in a vector a = vec(A)
of dimension L2 where the columns of A are stacked to form a. Observe that the symmetry restriction can be
expressed in a sigma notation (entry-wise) but this matrix algebra notation helps us to make more clear the use
of the trace operator in the objective function.
6
To complete the proof we let r(z) = −S σ (z)p be the feedback error (of non-singularity in
prices).9 Then we can write the matrix function W σ (z) =
r(z)p0 +pr(z)0
p0 p
−
[r(z)0 p]pp0
(p0 p)2
that is the
symmetric part of the outer product of the feedback error and the price vector. Replacing r(z)
by its definition, we get that W σ = −E π :
E π (z) =
1
1
σ
0
0 σ
0 σ
0
p0 p [S (z)pp + pp S (z) − p0 p pp S (z)pp ],
0
then E π = S σ − P S σ P with P = I − ppp0 p
and by Walras’ law it simplifies to:
E π (z) =
1
σ
0
p0 p [S (z)pp
+ pp0 S σ (z)],
this expression along with the implied multipliers λ and U , satisfies all the first-order conditions of the problem. Since we can use arguments identical to those in Claim 1 –only not
imposing NSD–, we know that the solution is unique. Hence, this expression describes the solution to the posed calculus of variations problem with the symmetry and singularity restrictions.
The proof is complete.
If S σ,π were NSD, then we would be done, since it would have property R and minimize
||E||2 . Otherwise, the general solution is provided after the following lemma, which rewrites the
problem slightly.
Lemma 2. The matrix nearness problem can be rewritten as minA ||S σ,π − A||2 subject to A ∈
M(Z) having property R.
Proof. Recall the matrix nearness problem: minA∈M(Z) ||S − A||2 subject to A satisfying singularity, symmetry, and NSD. This is equivalent, by manipulating the objective function, to:
minA∈M(Z) ||E σ + E π + S σ,π − A||2 . Writing out the norm as a function of the traces, and using
the fact that E σ is skew symmetric, while the rest of the expression is symmetric, we get that this
amounts to writing: minA∈M(Z) ||E σ ||2 +||E π +S σ,π −A||2 subject to A having property R. This
is in turn equivalent to: minA∈M(Z) ||E σ ||2 + ||E π ||2 + 2hE π , [S σ,π − A]i + ||S σ,π − A||2 subject to
A having property R. Then, exploiting the fact that E π and S+ = S σ,π − A are orthogonal (as
proved in Claim 3), we obtain that the problem is equivalent to minA∈M(Z) ||E σ ||2 + ||E π ||2 +
||E ν ||2 subject to A having property R.
Hence, since the objective function of the matrix nearness problem ||E||2 = ||E σ ||2 +||E π ||2 +
||S σ,π −A||2 , solving the program minA∈M(Z) ||E||2 subject to A having property R is equivalent
to solving minA∈M(Z) ||S σ,π − A||2 subject to the same constraints.
Now, the best NSD matrix approximation of the symmetric valued function S σ,π is S r =
S σ,π,ν . Then, the candidate solution to our problem is A(z) = S r (z) for all z ∈ Z. Notice that
S r (z) is symmetric and singular with p in its null space by construction. Indeed, recall that
S σ,π (z) = Q(z)Λ(z)Q(z)0 and S r (z) = S σ,π,ν (z) = Q(z)Λ(z)− Q(z)0 . Then it follows that S r (z)
is symmetric for z ∈ Z. Moreover, by definition λl (z)− = min(0, λl (z)) for l = 1, . . . , L. Since
S σ,π (z)p = 0, it follows that λL (z) = 0 is the eigenvalue function associated with the qL (z) = p
eigenvector. Then we have that λL (z)− = 0 is also associated to the eigenvector p, and we can
conclude that S r (z)p = 0.
As just argued, S r (z) has property R, i.e., S r (z) is in the constraint set of the matrix nearness
problem. We conclude that it is its solution.
The orthogonalities of the relevant subspaces of matrix functions, as made evident in previous
steps of the proof, are responsible for the additive decomposition of the norm in the second part
9 In
this step we follow a similar strategy of proof to Dennis & Schnabel (1979); Higham (1989).
7
of the statement. The Frobenius norm and its weighted variants are essentially the only ones
that correspond to a an inner product on the space of matrix valued functions. For this reason,
they are the only ones that lead to the additive decomposition of the three components of the
residual matrix E.
Furthermore, we close this section by proving that S r is a continuous matrix-valued function.
Indeed, the following is a property of the solution to our problem:
Claim 2. S r is continuous.
Proof. Note that S r is the result of three projections on closed subspaces applied to the convex
set of constraints. Such projections are continuous mappings under the conditions that we have
imposed, and then S r is continuous by construction in all z ∈ Z.
4
Decomposition of the Matrix Nearness Solution
The importance of Theorem (1) is to provide a precise quantification of the size of the departures
from rationality by a given behavior, as well as a revealing decomposition thereof.10
We should think of the three terms in the decomposition of ||E||2 as the size of the violation
of symmetry, the size of the violation of singularity, and the size of the violation of negative
semidefiniteness of a given Slutsky matrix, respectively (this idea is illustrated in figure 1). The
three terms are the antisymmetric part of the Slutsky matrix function, the correcting matrix
function needed to make the symmetric part of the Slutsky matrix function p-singular, and
the PSD part of the resulting corrected matrix function. Note that if one is considering a
rational consumer, the three terms are zero. Indeed, if S(z) satisfies property R for all z ∈ Z,
S(z) = S σ (z) and E σ (z) = 0, S σ,π (z) = S σ (z) and E π (z) = 0, and S r (z) = S σ,π,ν (z) = S σ,π (z)
and E ν (z) = 0. If exactly two out of the three terms are zero, the nonzero term allows us
roughly to quantify violations of the Ville axiom of revealed preference –VARP–, violations of
homogeneity of degree 0, or violations of the compensated law of demand (the latter being
equivalent to the weak axiom of revealed preference –WARP–), respectively. We elaborate on
these connections with the axioms of consumer theory in Subsection 4.1 below.
The violations of the property R have traditionally been treated separately. For instance,
Russell (1997), using a different approach (outer calculus), deals with violations of the symmetry
condition only. In this case, ||E||2 = ||E σ ||2 .
Another application of our result connects with Jerison and Jerison (1993), who study the
case of violations of symmetry and negative semidefiniteness independently. They prove that
the maximum eigenvalue of S σ (z) can be used to bound ||e||C locally when NSD is violated and
E σ (z) can be used to bound ||e||C when symmetry is violated. Indeed, this is consistent with
our solution. In this case ||E||2 = ||E σ ||2 + ||S σ + ||2 , where max({λ̃l (z)+ }l=1,...,L ) ≤ ||S σ + ||2 ≤
d · max({λ̃l (z)+ }l=1,··· ,L ), with d = Rank(S σ (z)+ ) (by the norm equivalence of the maximum
eigenvalue and the Frobenius norm).
4.1
Connecting with Axioms of Revealed Preference
Since Hurwicz and Uzawa (1971) it is known that for the class of continuous differentiable
functions a demand that satisfies Walras´ law can be rationalized if and only if its Slutsky
10 The
decomposition, a consequence of orthogonalities, can be seen as a generalized Pythagorean theorem.
8
S
ÈÈ E Σ ÈÈ 2
SΣ
ÈÈ EÈÈ 2 =ÈÈ E Σ ÈÈ 2 +ÈÈ E Π ÈÈ 2 +ÈÈ E Ν ÈÈ 2
ÈÈ E Π ÈÈ 2
ÈÈ E Ν ÈÈ 2
S Σ,Π
Sr
Figure 1: Decomposition of the Slutsky matrix norm. The size of the violations of rationality is
the sum of the violations of symmetry, singularity in prices and negative semidefiniteness.
matrix function is symmetric (σ) and NSD (ν). A corollary of this result is that its Slutsky matrix
function is singular in prices (π) (John, 1995). These Slutsky regularity conditions (σ, π, ν) are
linked to behavioral demand axioms. In fact, Houthakker (1950) proved that, for the class of
continuous demand functions, the strong axiom of revealed preference –SARP– implies that a
demand can be rationalized. In this sense, SARP is indeed strong. A weaker axiom implies
only the Slutsky matrix function symmetry condition: the Ville axiom of revealed preference
–VARP– is equivalent to the symmetry condition and therefore to integrability of the demand
system (Hurwicz & Richter, 1979). VARP postulates the nonexistence of a Ville cycle in the
income path of a demand function. The weak axiom of revealed preference –WARP– implies
that the Slutsky matrix function is NSD and singular in prices. Furthermore, the NSD and
singularity in prices are equivalent to a weak version of WARP (Kihlstrom et al., 1976). VARP
is a differential axiom and does not imply SARP or WARP (Hurwicz & Richter, 1979). It is also
known that WARP does not imply VARP or SARP for dimensions greater than two.
A continuously differentiable demand function is said to be rationalizable if and only if
it fulfills SARP. However, we can also combine the other weaker axioms to have the same
result while drawing connections to the additive components of our Slutsky norm. For example,
VARP and WARP imply that a demand function is rationalizable, and so do the Wald Axiom,
homogeneity of degree zero and VARP.
Before presenting the main remark of this subsection, for completeness, it is useful to posit
the axioms that we employ.
Assumed throughout, the first basic condition (price is a Slutsky matrix left eigenvector) is
given by Walras’ law.
Axiom 1. (Walras’ law) p0 x(p, w) = w.
We have that if x ∈ X (Z) satisfies Walras’ law then its Slutsky matrix function S ∈ M(Z)
has the following property: p0 S(z) = 0 for z ∈ Z. The converse holds when Z is path connected.
The condition that the price p is the Slutsky matrix right eigenvector or singularity in prices
is given by “no money illusion”.
9
Axiom 2. (Homogeneity of degree zero -HD0-) x(αz) = x(z) for all z ∈ Z and α > 0.
It is easy to prove that if x ∈ X (Z) satisfies Walras’ law and HD0 then S(z)p = 0 for z ∈ Z.
More generally, p is an eigenvector associated with the null eigenvalue of S if and only if x
satisfies Walras’ law and HD0.
The symmetry of the Slutsky matrix function is given by VARP.11 To state this axiom we
need to define an income path as w : [0, b] 7→ W and a price path p : [0, b] 7→ P . Let (w(t), p(t))
be a piecewise continuously differentiable path in Z. Jerison and Jerison (1992) define a rising
∂p
real income situation whenever ( ∂w
∂t (t), ∂t (t)) exists, leading to
Ville cycle is a path such that: (i) (w(0), p(0)) = (w(b), p(b)); and
∂p
∂w
0
∂t (t) > ∂t (t) x(p(t), w(t)). A
∂p
0
(ii) ∂w
∂t (t) > ∂t (t) x(p(t), w(t))
for t ∈ [0, b].
Axiom 3. (Ville axiom of revealed preference -VARP-) There are no Ville cycles.
Hurwicz and Richter (1979) proved that x ∈ X (Z) satisfies VARP if and only if S is symmetric
and x satisfies Walras’ law.
The negative semidefiniteness condition of the Slutsky matrix is given by the Wald axiom.
The Wald axiom itself is imposed on the conditional demand for a given level of wealth. Following
John (1995) a demand function that satisfies Walras’ law is said to fulfill the Wald axiom
when so do the whole parametrized family (for w ∈ W ) of conditional demands. Formally, a
demand function can be expressed as the parametrized family of conditional demands. That is:
x(p, w) = {xw (p)}w∈W where xw : P 7→ X such that p0 xw (p) = w for all p ∈ P .
Axiom 4. (Wald axiom) x ∈ X (Z) is such that for every w ∈ W and for all p and p, p0 xw (p) ≤ w
=⇒ p0 xw (p) ≥ w.
The Wald axiom implies that S ≤ 0 (John, 1995) for demands that satisfy Walras’ law.
The Slutsky singularity in prices and the NSD conditions are equivalent to the following
version of WARP.
Axiom 5. (Weak axiom of revealed preference -WARP-) If for any z = (p, w) z = (p, w):
p0 x(p, w) ≤ w =⇒ p0 x(p, w) ≥ w.
This is the weak version of WARP, as in Kihlstrom et al. (1976). We follow John (1995),
who proves that for continuously differentiable demands (that satisfy Walras’ law) WARP is
equivalent to the Wald Axiom and HD0. Kihlstrom et al. (1976) and John (1995) prove that
x ∈ X (Z) satisfies WARP if and only if S is NSD and S(z)p = 0.
We are ready to summarize the main point of this subsection in the following remark.
Remark 1. The Slutsky matrix nearness norm ||E||2 will be equal to zero if and only if x satisfies
VARP, homogeneity of degree zero, and the Wald Axiom (in addition to Walras’ law).
Moreover,
(i) ||E||2 = ||E σ ||2 > 0 only if VARP fails. In this case, either HD0 and the Wald Axiom
hold, or Warp holds.
(ii) ||E||2 = ||E π ||2 > 0 only if Walras’ law fails. In addition, HD0 may also fail; VARP and
the Wald axiom hold.
(iii) ||E||2 = ||E ν ||2 > 0 only if the Wald axiom fails. In this case, VARP holds, and either
WARP fails or HD0 holds.
11 We present the axiomatization due to Ville as reinterpreted by Hurwicz and Richter (1979) and Jerison and
Jerison (1992). There are alternative discrete axioms due to Jerison and Jerison (1996), that also do the job and
are potentially testable.
10
5
Weighted Slutsky Norms
The norm of bounded rationality that we have built so far is an absolute measure. Therefore,
for a specific consumer, this distance quantifies by how far that individual’s behavior is from
being rational. Furthermore, one also can compute how far two or more consumers within a
certain class are from rationality, and induce an order of who is closer in behavior to a rational
consumer. However, we are limited to the case where the setting of the decision making process
is fixed in the sense that the decision problem facing each of the individuals is presented in the
same way. This implies that the measure is unit dependent, being stated in the same units
(the units in which the consumption goods are expressed). A second undesirable feature, is that
the Frobenius norm depends on the area of the arbitrary compact region Z that has positive
Lebesgue measure. This section addresses these issues.
Let W , U be square and PSD (i.e., with symmetric part also PSD) matrix functions of
weights such that it is non-null and W (z), U (z) ∈ RL × RL . This matrix pair is meant to encode
all suitable normalizations and priorities of the modeller. Let Z ⊆ P × W be sigma-measurable
and have positive measure µ with µ continuous, strictly positive function, such that S ∈ M(Z)
is square integrable. Let ||·||M be the Frobenius matrix norm where M = RL ×RL . The weighted
semi-norm ||S||W U,µ is defined as follows:
ˆ
||S||2W U,µ =
z∈Z
||W (z)S(z)U (z)||2M µ(z)dz < ∞.
When µ is continuous and strictly positive, the spaces L2 (Z, B(Z); M(Z)) and the weighted
L2 (Z, B(Z); M(Z); µ) are unitary equivalent or unitary isomorphic. This means that our orthogonality relations under the inner products defined by the Lebesgue measure are preserved
by inner product weighted by µ. Hence, suitable variants of Theorem 1 can be obtained for
these weighted norms.
Remark 2. With a W = U that is positive definite and symmetric, the solution to the matrix
nearness problem, S ∗r = argminA∈M(Z) ||S − A||W,µ subject to A having the property R, is
S ∗r = W −1 (W S σ,π W )− W −1 and the quadratic norm is given by:
||E ∗ ||2W,µ = ||E ∗σ ||2W,µ + ||E ∗π ||W,µ + ||E ∗ν ||2W,µ ,
where E ∗σ = E σ , E ∗π = E π , and E ∗ν = W −1 (W S σ,π W )− W −1 .
Observe that only the part concerning the NSD matrix function of violations (ν property)
changes due to the introduction of the weighting matrix function W .
Remark 3. Using the identity matrix as W = U and the multivariate uniform as µ, one obtains
the closest results to Theorem 1. However, the resulting Slutsky norm is independent of the area
of the domain Z.
Remark 4. If W = Diag(p) and U = W , the weighted Slutsky matrix is expressed in (square)
dollar amounts. The entries of this normalized matrix express changes in expenditure on each
good following a percent change in each price.
Remark 5. If W =
√1 Diag(p)
w
and U = W , the weighted Slutsky matrix is unit-free. The
entries of this normalized matrix express changes in expenditure shares on each good following
a percent change in each price.
11
Remark 6. When we have a degenerate µ such that µ(z) = 1 (say an equilibrium price and a
given exogenous wealth) and W, U are the identity, ||S||µ = ||S(z)||M and we can apply the local
results of Jerison and Jerison (1992).
The standard Frobenius norm we use is chosen when one gives equal weight to each entry of
the Slutsky matrix, thus avoiding that a specific price change on a specific commodity receive
greater weight, but the modeller may adopt such choices at will and use arbitrary weighting
matrices as a function of her interests. Finally, the integral in the norm embodies an expectation
operator, which can be justified with Savage-type ideas of independence across consumption data
points.
6
The Comparative Statics of Bounded Rationality
An approach based on the Slutsky matrix cannot be silent on comparative statics analysis.
This section goes over a number of results that shed light on the meaning of the decomposition
found in Theorem 1 and that connect with the behavioral nearness problem in the space of
demand functions. The first subsection below focuses on mis-specification errors, offering another
result based on orthogonalities. The results in the second and third subsections relate errors in
comparative statics predictions to the error terms in our Slutsky norm decomposition.
6.1
Connecting with Demand: Mis-Specification Errors
We have solved the matrix nearness problem on the basis of the Slutsky regularity conditions.
Now, in order to connect with demand, the exercise is one of integrating from the first-order
derivatives of the Slutsky matrix terms. In such an integration step, a constant of integration
shows up, which we would like to envision as a “residual” or “mis-specification error.” That
is, starting from our observed Slutsky matrix function S(x), we find thanks to Theorem 1 the
nearest matrix function S r satisfying all the regularity properties. Now suppose that we specify
x1 and x2 as two demand functions with which we wish to explain the consumer’s behavior
(for example, x1 might be a Cobb-Douglas demand properly estimated, whereas x2 might be a
different estimate, perhaps allowing the entire class of CES demands).
As will be shown, the error in comparative statics analysis from using each of those specifications is, respectively, ||S(x) − S(x1 )||2 = ||S r − S(x1 )||2 + ||E||2 and ||S(x) − S(x2 )||2 =
||S r − S(x2 )||2 + ||E||2 . This means that ||E||2 = ||S(x) − S r ||2 found in Theorem 1 –the
behavioral error– does not change with x1 or x2 , this term is invariant to the choice of the
parametrized rational approximation. The other term is the mis-specification error, as a function of the parametrized class considered. This is a useful separation. We establish it in our
next result.
r
Let R (Z) ⊂ R(Z) be an arbitrary closed and bounded subset of rational demand functions
in X (Z).12 In particular, we are interested in a symmetric error, where we penalize any mistake
symmetrically which corresponds to the Frobenius matrix norm:
minxr ∈Rr (Z) ||S(xr ) − S(x)||2 .
12 Lacking compactness, replace the min with the inf operator in the problem below. In fact, R(Z) is not
compact but the result goes through.
12
Our next result follows:
Proposition 1. For any solution x∗r ∈ argminxr ∈Rr (Z) ||S(xr )−S(x)||2 , it follows that ||S(x∗r )−
S(x)||2 = ||S(x∗r ) − S r ||2 + ||E||2 , where ||S(x∗r ) − S r ||2 is a miss-specification error and ||E||2
is the Slutsky matrix norm.
Proof. The proof has three steps: First, we use the decomposition of a matrix into its symmetric
and antisymmetric parts: ||S(x) − S(xr )||2 = ||S − S(xr )||2 = ||S σ + E σ − S(xr )||2 = ||S σ −
S(xr )||2 + 2hE σ , S σ − S(xr )i + ||E σ ||2 ,
where S σ −S(xr ) is a symmetric matrix function and E σ is a skew-symmetric matrix function.
Then hE σ , S σ − S(xr )i = 0. Then ||S(x) − S(xr )||2 = ||S σ − S(xr )||2 + ||E σ ||2 .
Second, we take the p-singularity projection: ||S σ − S(xr )||2 = ||S σ,π + E π − S(xr )||2 =
||S σ,π − S(xr )||2 + 2hE π , S σ − S(xr )i + ||E π ||2 . Since E π is orthogonal to the matrix functions
that are symmetric and have p as its eigenvector associated with the null eigenvalue (claim 3),
by definition S(xr (p, w))p = 0 and S(xr ) is symmetric. Then: hE π , S σ,π − S(xr )i = hE π , S σ,π i −
hE π , S(xr )i = 0. Therefore, ||S(x) − S(xr )||2 = ||S σ,π − S(xr )||2 + ||E σ ||2 + ||E π ||2 .
Finally, we deal with the NSD part: ||S σ,π − S(xr )||2 = ||S σ,π,ν + E ν − S(xr )||2 = ||S σ,π,ν −
S(xr )||2 +2hE ν , S σ,π,ν −S(xr )i+||E ν ||2 . Using the vector space structure of the space of demand
functions, we can write e = x − xr . Then S(x) = S(xr ) + G, where G = Dp e + Dw [xe0 ] + Dw [ee0 ].
It follows that: hE ν , S σ,π,ν − S(xr )i = hE ν , Gσ,π,ν i = 0, where the right hand side follows from
the orthogonality of E ν and Gσ,π,ν . This implies: ||S(x) − S(xr )||2 = ||S r − S(xr )||2 + ||E||2 .
In the same spirit, we close this subsection with a local result. Around a given reference price
p, if one searches for a minimizer of error in comparative statics with respect to S r among all
twice continuously differentiable functions with bounded Hessians, satisfying Walras’ law and
defined on an open ball around p, the result is a rational demand function. Note that the space
of demands that satisfies Walras’ law is convex.
Define the local problem. Fix a reference price p. We can compute the Slutsky matrix of x
at p, S(p, w) where we allow w to vary. We can compute S r (p, w) using the same techniques
developed for the general case. Abusing notation we keep referring to S(p, ·) as S and S r (p, ·) as
S r . The errors in comparative statics made due to assuming (perhaps incorrectly) rationality,
around an open ball centered at p correspond to the problem:
miny∈X (p) ||S(y) − S r ||2
where X (p) is the set of demands that fulfill Walras’ law that are defined in an open ball
around p, and that are twice continuously differentiable with a bounded Hessian. Let R(p) ⊆
X (p) be the subset of rational demands with the same properties. This is a local errors problem.
Proposition 2. If y ∗ ∈ argminy∈X (p) ||S(y) − S r ||2 , then y ∗ ∈ R(p).
Proof. The proof is organized in the following series of lemmata. We start with a definition due
to Jerison and Jerison (1992; 1993).
L
Let (p̂, ŵ) ∈ RL+1
++ , and let a price path be defined p(t) = tp+(1−t)p̂, where p ∈ R++ is fixed.
Then we study the Slutsky wealth compensation problem:
∂w(p̂,ŵ,p,t)
∂t
=
∂p(t)
∂t x(p(t), w(p̂, ŵ, p, t))
∗
with initial condition w(p̂, ŵ, p, 0) = ŵ. The unique solution to this problem is w (p̂, ŵ, p, ·).
The following definition of approximate local indirect utility is borrowed from Jerison and
Jerison (1992; 1993). The indirect local utility at (p̂, ŵ), where p̂ is close enough to p ( is
13
defined by v σ (p̂, ŵ) = w∗ (p̂, ŵ, p, 1). Jerison and Jerison (1993) prove that w∗ is continuously
differentiable when x is continuously differentiable, and it is known that continuous functions
can be approximated arbitrarily closely (in the sup norm) by smooth functions (Azagra & Boiso,
2004), thus we assume that w∗ is smooth.
Lemma 3. [Jerison and Jerison (1993)] For every demand x ∈ X (p) that is twice continuously
differentiable there exists xσ ∈ X (p) such that S(xσ )(p, w) = S σ (p, w). Moreover, xσ (p, w) =
∇ v σ (p,w)
− ∇wp vσ (p,w) .
Proof. Remark 1 of [Jerison and Jerison (1993)].
Lemma 4. If y ∈ argminy∈X (p) ||S(y) − S r ||2 then S(y) is symmetric. Moreover, if Walras’ law
holds and S σ,π = S r there is a y ∗ ∈ argminy∈X (p) ||S(y) − S r ||2 such that y ∗ = xσ .
Proof. Take y ∗ ∈ argminy∈X (p) ||S(y) − S r ||2 and assume that S(y ∗ ) is not symmetric. Then let
∗
S(y ∗ ) = S y . Now we can compute ||S(y ∗ ) − S r ||2 = ||S y
||S y
S
∗
y ,σ
∗
,σ
+ Ey
∗
,σ
− S r ||2 = ||S y
r
∗
,σ
−S is symmetric, we have ||S
− S r ||2 + 2hE y
∗
y ,σ
+E
∗
y ,σ
∗
,σ
r 2
, Sy
∗
∗
,σ
,σ
−S || = ||S
+ Ey
∗
,σ
− S r ||2 . Observe that:
− S r i + ||E y
∗
y ,σ
∗
r 2
,σ 2
−S || +||E
(3) there exists y ∗σ ∈ X (p) for all y ∗ such that ||S(y ∗σ )−S r ||2 = ||S y
∗
,σ
|| . By the fact that
y ∗ ,σ 2
|| . By the lemma
−S r ||2 . This means that
||S(y ∗ ) − S r ||2 > ||s(y ∗σ ) − S r ||2 which is a contradiction to y ∗ ∈ argminy∈X (p) ||S(y) − S r ||2 .
We conclude that S(y ∗ ) is symmetric.
The moreover part of the statement follows from the existence of xσ such that S(xσ ) = S x,σ
locally. By Walras’ law and symmetry, S x,σ (p, w)p = 0 thus xσ is HD0 around (p, w), this means
that locally S(xσ ) = S x,σ,π . Thus ||S(xσ ) − S r ||2 = 0 when S x,σ,π = S r around (p, w), thus
making xσ a solution.
Lemma 5. [CCD] Convex concave decomposition. Any scalar twice continuously differentiable
function with a bounded Hessian can be decomposed in the sum of a concave and convex function
parts.
Proof. Yuille & Rangarajan (2003)
Our task now is to define a local expenditure function. For a fixed w, let eσ (p, u) be defined
implicitly as v σ (p, eσ (p, u)) = u. The eσ is not necessarily concave in p. The function eσ is
unique and continuous when the indirect utility is monotone in wealth.13
Now notice that, by construction, the integrable demand xσ can be written as xσ (p, e(p, u)) =
∇p eσ (p, u) because by differentiating v σ (p, eσ (p, u)) = u with respect to prices we obtain:
∇ v σ (p,eσ (p,u))
∇p eσ (p, u) = − ∇wp vσ (p,eσ (p,u)) . In turn, the Slutsky matrix of a “integrable” demand xσ can be
written as: S(xσ )(p, e(p, u)) = Heσ (p, u), where H stands for the Hessian operator.
Lemma 6. If y ∈ argminy∈X (p) ||S(y) − S r ||2 , then S(y) is NSD.
13 If it were not monotone, in particular if existence or uniqueness are not guaranteed (i.e., the problem is
ill-posed) then we could use a regularization procedure to obtain a unique continuous expenditure (Tichonov
regularization). However, the result goes through also if there exists at least one continuous and differentiable
solution eσ to the equation above so we do not need uniqueness. Thus without loss of generality we assume that
v σ (p, w) is monotone in wealth.
14
Proof. By lemma (3) we know that y ∗ = y ∗σ or equivalently S(y ∗ ) is symmetric. Also we
know that there is a local quasi-expenditure function eσ (p, v(p, w)) = w for a fixed w such that
S(y ∗ )(p, eσ (p, u)) = He(pσ , u). Now take the matrix function S r and evaluate it at the same
local quasi-expenditure function and obtain the matrix function: H x,r (p, u) = S r (p, eσ (p, u)).
Next, decompose the quasi-expenditure function in a concave part eσcave = eσ − r with r a
convex function and a convex part eσvex = r. We focus on the local p on quadratic expenditure
functions (this is without loss of generality locally due to the Taylor expansion under twice
continuous differentiability). Then we can ensure that we can find the minimal decomposition
in the sense of Frobenius such that Heσ = Heσcave + Heσvex such thathHeσcave , Heσvex i = 0 and
Heσcave is NSD and Heσvex is PSD (because locally we can think of the allowable functions as
approximately quadratic forms). We study the equivalent problem ||S(y ∗ ) − S r ||2 = ||Heσ −
H x,r ||2 .
Assume by contradiction that y ∗ (a solution to the problem) is such that S(y ∗ ) is not NSD.
σ
Observe that eσ associated with y ∗ provides the value ||Heσ − H x,r ||2 = ||Heσcave + Hvex
−
H x,r ||2 . This in turn can be expanded ||Heσcave + Heσvex − H x,r ||2 = ||Heσcave − H x,r ||2 +
σ
2hHeσvex , Heσcave − H x,r i + ||Heσvex ||2 . By construction, 2hHeσcave , Hvex
i = 0. Then ||Heσcave −
H x,r ||2 + 2hHeσvex , −H x,r i + ||Heσvex ||2 > ||Heσcave − H x,r ||2 because hHeσvex , −H x,r i ≥ 0, because
both matrices are PSD. This contradicts that y ∗ is a solution. Therefore, y ∗ is such that S(y) is
symmetric and NSD.
Lemma 7. If Walras’ law holds and if y ∗ ∈ argminy∈X (p) ||S(y) − S r ||2 , then S(y ∗ )(p, w)p = 0
( singular in p). Moreover, y ∗ is HD0.
Proof. Under Walras’ law and by the results of lemma (6) we apply John (1995) result to conclude
that S(y ∗ )(p, w)p = 0 so it is singular in p. By Walras’ law again, this leads us to conclude that
Dp y ∗ (p, w)p + Dw y ∗ (p, w)w = 0, this implies that y ∗ is HD0.
This last result also yields an explicit solution locally. Observe that by definition eσ (p, v(p, w)) =
w.
Definition 4. [Concave part local indirect utility] We define the “concave part” of the indirect
utility vcave implicitly eσcave (p, vcave (p, w)) = w.
We also define a notion of Slutsky matrix function integrability:
Definition 5. [Slutsky integrability] We say a matrix function S ∈ M(Z) is Slutsky integrable
if there exists some x ∈ X (Z) such that S(x) = S.
This definition can be modified appropriately for the local setup:
Corollary 1. The closest rational demand to x ∈ X (p) in the sense of minimizing its compar∇ v
(p,w)
. Moreover
ative statics error is given by xr ∈ R(p) defined locally as xr (p, w) = − ∇wp vcave
cave (p,w)
r
2
ι 2
its value function can be decomposed in four components ||S(x ) − S(x)|| = ||E || + ||E σ ||2 +
||E π ||2 + ||E ν ||2 , where ||E ι ||2 = miny∈X (p) ||S(y) − S r ||2 is the mis-specification error (distance
of S r from being the Slutsky matrix of some demand in X (p), i.e., Slutsky integrability).
Proof. This follows from Proposition 2 and Theorem 1.
15
6.2
Path Independence of Wealth Compensations
In what follows we study the Slutsky Wealth compensation problem and its relation with the
Slutsky Error E σ . In this and the next subsection we assume that the price-wealth region Z is
path connected. We need some preliminary definitions.
Definition 6. A price path is p : [0, 1] 7→ RL
++ with p(0) = p0 and p(1) = p1 is a continuous
function that connects two points in a convex subset of RL
++ .
The Slutsky wealth compensation problem is that of compensating a given price change
through a change in wealth in order to keep the initial demand quantities affordable after the
change in prices. We ask what is the local wealth compensation needed to offset a small price
change at any point in the price path. We want to compute the total Slutsky wealth compensation
that is required to move from price situation p0 to p1 . This means that along the path the
consumer is always compensated in the sense of Slutsky for small price changes.
The local Slutsky compensation quantity change is related to the actual amount or level of
wealth compensation by the following relation :
∂w(t)
∂t
=
∂p(t) 0
∂t x(p(t), w(t))
with Walras’ law
holding along the path w(t) = p(t)0 x(t), in particular we fix w(0) = w0 at some level. Hence, the
´t
)0
Slutsky compensation level at some t is given by w(t) = w(0) + 0 ∂p(τ
∂τ x(p(τ ), w(τ ))dτ along
a price path p. Of course, the sum total wealth compensation change from (p0 , w0 ) to p1 is:
´1
)0
w(1) − w(0) = 0 ∂p(τ
∂τ x(p(τ ), w(τ ))dτ .
We next show that E σ is related to a notion of path independence in this wealth compensation. To introduce the path independence definition, we define a composite price path.
Definition 7. (Composite price path) Let p and p∗ be price paths such that p(0) = p∗ (0) and
p(1) = p∗ (1). A composite price path is defined as p(s, t) = ps (t) = sp(t) + (1 − s)p∗ (t) for all
s ∈ [0, 1] and all t ∈ [0, 1].
To define path independence, we first need to consider the Slutsky wealth compensation prob´ t s (τ )
0
s (t)
s (t)
= ∂p∂t
x(ps (t), ws (t)) with solution ws (t) = ws (0)+ 0 ∂p∂τ
x(ps (τ ), ws (τ ))dτ .
lem for a fixed s: ∂w∂t
Next, we need to allow s to vary in order to capture the sensitivity to the paths. This leads to
the following definition:
Definition 8. (Slutsky Wealth Compensation within-Path gap) The Slutsky Wealth Compensation within-Path gap is w0 (1) − w1 (1).
In this exercise, we have allowed τ first to trace the interval [0, 1], followed by a similar move
in the parameter s.
Alternatively, we can allow first s to move, followed by a move in τ in determining the wealth
compensations. To study first the relation of the dependence of the wealth compensation on the
path (parameter s), we propose the following quantity (whose exact expression stems from the
integral calculations in the ensuing proof). This is well defined for any composite price and
wealth paths ps and ws .
Definition 9. (Uncompensated across-path demand correction term) The uncompensated across´ 1 ´ 1 s (τ ) 0
s (τ )
[Dw x(ps (τ ), ws (τ ))ps (τ )0 ] ∂x∂s
dsdτ , (i.e, the
path demand correction term c = − 21 0 0 ∂p∂s
0
s ∂xs
weighted inner product h ∂p
∂s , ∂s iAs (τ ) = 0 where As (τ ) = Dw x(ps (τ ), ws (τ ))ps (τ ) .
16
We refer to c as uncompensated across-path demand correction term because in general
changes in wealth are not required to be compensated with respect to the parameter s, which is
first allowed to move, and c captures this divergence.
When the ps and ws are not Slutsky compensated paths with respect to s, the c correction
term allows us to obtain an equivalence between a quadratic form of E σ and a monetary measure
of the possible departures of rationality. We propose the following definition, in which the order
of moving s or τ is irrelevant in affecting the wealth compensations:
Definition 10. (Slutsky wealth compensation path independence) Slutsky wealth compensation
path independence holds whenever
[w0 (1) − w1 (1)] − c = 0.
Our next result follows:
Proposition 3. The Slutsky Wealth Compensation is path independent if and only if E σ = 0.
Moreover, for any price ps and wealth ws paths, the corrected path dependence wealth gap
1
[w1 (1) − w0 (1)] − c =
2
ˆ
0
1
ˆ
0
1
0
∂ps (τ ) σ
∂ps (τ )
dτ ds.
E (ps (τ ), ws (τ ))
∂s
∂τ
Proof. The first part of the statement follows from the Frobenius-Stokes theorem of integrability
(Guggenheimer, 1962).
For the “moreover” part of the statement, given s, we consider the function ws (t), the solution
´ t s (τ ) 0
x(ps (τ ), ws (τ ))dτ .
for that s of the Slutsky Wealth Compensation problem: ws (t) = ws (0)+ 0 ∂p∂τ
We compute the derivative with respect to s,
∂ws (t)
=
∂s
ˆ
0
t
∂ws (t) 14
∂s :
0
∂ps (τ )
∂ps (τ )
dτ +
Dp x(ps (τ ), ws (τ ))
∂τ
∂s
ˆ
0
t
0
∂ps (τ )
∂ws (τ )
dτ +
Dw x(ps (τ ), ws (τ ))
∂τ
∂s
ˆ t 2
0
∂ ps (τ )
x(ps (τ ), ws (τ ))dτ.
∂s∂τ
0
Focusing on ws (1) and using integration by parts on the third term:
ˆ
0
1
ˆ 1
0
0
0
∂ 2 ps (τ )
∂ps (τ )
∂ps (τ )
∂ps (τ )
x(ps (τ ), ws (τ ))dτ =
+
x(τ )|10 − [
[Dp x(ps (τ ), ws (τ ))]
∂s∂τ
∂s
∂s
∂τ
0
ˆ 1
0
∂ps (τ )
∂ws (τ )
[Dw x(ps (τ ), ws (τ ))]
dτ ].
∂s
∂τ
0
Notice that
Thus:
14 This
∂ps (τ ) 0
x(τ )|10
∂s
= p∗ (τ )0 x(τ )|t0 − p(τ )0 x(τ )|t0 . By Walras’ law this term vanishes.
function is differentiable (by the results given by Guggenheimer (1962)).
17
ˆ
∂ws (1)
=
∂s
1
0
ˆ
0
1
∂ps (τ )
∂ps (τ )
[Dp x(ps (τ ), ws (τ )) − Dp x(ps (τ ), ws (τ ))0 ]
dτ +
∂τ
∂s
0
ˆ
0
0
1
∂ps (τ )
∂ps (τ )
∂ws (τ )
∂ws (τ )
Dw x(ps (τ ), ws (τ ))
dτ −
[Dw x(ps (τ ), ws (τ ))]
dτ.
∂τ
∂s
∂s
∂τ
0
By definition of the price path ps and by Walras’ law, we have ws (τ ) = ps (τ )0 xs (τ ). This
implies:
∂ws (τ )
=
∂s
s (τ )
x(τ )0 ∂p∂τ
.
s (τ )
xs (τ )0 ∂p∂s
+
∂xs (τ ) 0
ps (τ )
∂s
and by the Slutsky compensation equation
∂ws (τ )
∂τ
=
Now, plugging this inside the integral equation above:
ˆ
0
1
ˆ 1
0
0
∂ps (τ )
∂ps (τ )
∂ws (τ )
∂ws (τ )
Dw x(ps (τ ), ws (τ ))
dτ −
[Dw x(ps (τ ), ws (τ ))]
dτ =
∂τ
∂s
∂s
∂τ
0
ˆ 1
0
∂ps (τ )
∂ps (τ )
[Dw x(ps (τ ), ws (τ ))x(τ )0 − x(τ )Dw x(ps (τ ), ws (τ ))0 ]
dτ +
∂τ
∂s
0
ˆ 1
0
0
∂ps (τ )
∂xs (τ )
−
[Dw x(ps (τ ), ws (τ ))][
ps (τ )]dτ.
∂s
∂s
0
Therefore,
´ 1 s (τ ) 0 σ
s (τ )
= 2 0 ∂p∂τ
[E (ps (τ ), ws (τ ))] ∂p∂s
dτ + c(s) with
´ 1 ∂ps (τ ) 0
∂xs (τ ) 0
c(s) = − 0 ∂s [Dw x(ps (τ ), ws (τ ))][ ∂s ps (τ )]dτ, and
∂ws (1)
∂s
E σ (ps (τ ), ws (τ )) = 12 [S(ps (τ ), ws (τ )) − S(ps (τ ), ws (τ ))0 ].
Finally, by integration:
´1
´ 1 s (1)
ds + 0 c(s)ds
w1 (1) − w0 (1) = 0 ∂w∂s
´ 1 ´ 1 s (τ ) 0 σ
w1 (1)−w0 (1)
s (τ )
E (ps (τ ), ws (τ )) ∂p∂s
− c = 0 0 ∂p∂τ
dτ ds.
2
Next, we establish the connection with the Slutsky matrix norm. The local change of wealth
2
ws (t)
compensation with respect to the change of path is measured by ∂ ∂t∂s
− cs (t), and this term
is controlled globally by a weighted norm of E σ –suitable marginal quantity changes correcting
for possible path dependence– properly weighted by the changes in prices. Specifically:
Proposition 4. Given a composite price path, for any Slutsky wealth compensated path such
2
ws (t)
that ws (τ ) = ps (τ )0 x(τ ), it is the case that [ ∂ ∂t∂s
− cs (t)]2 = ||E σ (s, t)||2W U with || · ||W U the
√ ∂ps (t) ∂ps (t) 0
weighted Frobenius norm with weighting Ws (t) = 2 ∂s
when W is PSD pointwise and
∂t
U is the identity. Moreover, the path dependence wealth gap from any two paths is bounded above
by the average weighted Frobenius norm: [ 21 [w1 (1) − w0 (1)] − c]2 ≤ ||E σ ||2W U .
Proof. Observe that
∂ 2 ws (t)
∂t∂s
Note that we can write [
0
s (τ )
s (τ )
− cs (t) = 2 ∂p∂s
E σ (ps (τ ), ws (τ )) ∂p∂τ
.
∂ 2 wsx (t)
∂t∂s
− cs (t)]2 = T r([
∂ 2 wsx (t) ∂ 2 wsx (t)
∂t∂s ][ ∂t∂s ]),
where the left hand side is
a scalar and thus its trace is itself.
∂ 2 wsx (t) ∂ 2 wsx (t)
∂t∂s ][ ∂t∂s ]) =
0
0
s (t)
s (t) ∂ps (t)
s (t)
2T r( ∂p∂t
E σ (ps (t), ws (t)) ∂p∂s
E σ (ps (t), ws (t)) ∂p∂t
).
∂t
T r([
By the properties of the trace,
T r([
∂ 2 wsx (t)
∂t∂s
x,σ
2T r(2E
∂ 2 wsx (t)
∂t∂s − cs (t)]) =
0
0
s (t) ∂ps (t)
s (t) ∂ps (t)
(ps (t), ws (t)) ∂p∂t
E x,σ (ps (t), ws (t)) ∂p∂s
).
∂s
∂t
− cs (t)][
18
Then we can establish the first part of the result:
√
0
∂ 2 wsx (t)
∂ 2 wsx (t)
s (t) ∂ps (t)
− cs (t)]) = ||E x,σ (ps (t), ws (t)) 2 ∂p∂t
||2M
∂t∂s − cs (t)][
∂t∂s
∂s
√
0
s (t) ∂ps (t)
||E x,σ (ps (t), ws (t)) 2 ∂p∂t
||2M = ||Ws (t)E x,σ (ps (t), ws (t))||2M .
∂s
T r([
and
Finally, for the second part of the statement we apply Holder’s inequality to conclude that:
´ 1 ´ 1 ∂ps (τ ) σ
s (τ ) 2
[ ∂τ E (ps (τ ), ws (τ )) ∂p∂s
] dτ ds.
0 0
´ 1 ´ 1 ∂ps (τ ) σ
s (τ ) 2
By the first part of the statement, we conclude: 0 0 [ ∂τ E (ps (τ ), ws (τ )) ∂p∂s
] dτ ds =
´1´1
σ
2
σ 2
||Ws (t)E (ps (t), ws (t))||M dτ ds = ||E ||W U .
0 0
[ 21 [w1 (1) − w0 (1)] − c]2 ≤
6.3
The Compensated Law of Demand
So far we have related our error Slutsky matrix E σ to a path independence notion in wealth
compensated paths. Now we move on to relate E π and E ν to WARP and the Compensated law
of demand.
The compensated law of demand states that prices and quantities “move in opposite direction,” once the consumer’s wealth is compensated so that she can afford previous bundles. The
∂x(t)
compensated law of demand can be written in terms of price paths as ∂p(t)
≤ 0 for all
∂t
∂t
t ∈ [0, 1], for every Slutsky Wealth Compensated path. A Slutsky Wealth Compensation path
0 ∂p(t)
(p(t), w(t))t∈[0,1] is such that: ∂w(t)
∂t = x(p(t), w(t)) ∂t for all t ∈ [0, 1] .
Here we are interested in the sign and magnitude of the term
∂w(t)
∂t
∂p(t) 0 ∂x(p(t),w(t))
∂t
∂t
for Slutsky
x(p(t), w(t))0 ∂p(t)
∂t ,
Wealth Compensated paths
=
which we call locally compensated demand paths. The consumer that does not satisfy the compensated law of demand will have
∂p(t) 0 ∂x(t)
∂t
∂t
> 0 for some paths.
It is well known that the locally compensated demand is related to a quadratic form of
the Slutsky matrix function:
∂p(t) 0 ∂x(t)
∂t
∂t
=
∂p(t)
∂p(t) 0
∂t S(p(t), w(t)) ∂t .
We note that the locally
compensated demand path is measured in money.
Our next result relates the locally compensated law of demand to E π and E ν .
Proposition 5. Consider any locally compensated path. The locally compensated demand path
satisfies the compensated law of demand if and only if E π = 0 and E ν = 0.
∂p(t) 0 ∂x(t)
∂t
∂(t)
=
∂p(t) 0 r
∂t [S
π
+E +
Proof. We start by stating the definition
such that
r
∂w(t)
∂t
σ
Moreover,
E ν ] ∂p(t)
∂t
= x(p(t), w(t))0 ∂p(t)
∂t .
π
ν
∂p(t) 0 ∂x(t)
∂t
∂(t)
=
∂p(t) 0
∂p(t)
∂t S(p(t), w(t)) ∂t
for (p(t), w(t))t∈[0,1]
By virtue of the decomposition in Theorem 1, we can write
S = S + E + E + E . Also, we notice that
∂p(t) 0 σ
∂p(t)
∂t E (t) ∂t
= 0 for all t ∈ [0, 1] and any price
σ
path because E is a skew-symmetric matrix function and it vanishes in a quadratic form.
0 ∂x(t)
This establishes the second statement in the proposition. That is, this implies that: ∂p(t)
∂t
∂(t) =
∂p(t) 0 r
∂t [S (t)
+ E π (t) + E ν (t)] ∂p(t)
∂t .
Now, we turn to the first statement. If the compensated law of demand holds for any price
path and Slutsky wealth compensated path, it follows that:
σ
that S (t) is NSD so S
σ,π
∂p(t) 0 σ
∂p(t)
∂t [S (t)] ∂t
≤ 0 which implies
ν
is NSD and E = 0 must be zero by definition. This also implies that
S is NSD because so is its symmetric part, and by John (1995) (theorem 3) this implies that x
is HD0 which in turn implies that E π = 0.
Now, if E π = 0 and E ν = 0 then we conclude that S = S r + E σ . This implies in turn
0
∂p(t) 0 ∂x(t)
∂p(t)
r
= ∂p(t)
because E σ is skew-symmetric. By construction,
∂t
∂t
∂t S (p(t), w(t)) ∂t
0
∂p(t)
∂p(t)
r
∂t S (p(t), w(t)) ∂t ≤ 0 so the compensated law of demand holds for any locally compensated
that
demand path.
19
The dot product of price and quantity changes defined above is important. Its discretized
version (pt − pt−1 )0 (xt − xt−1 ) ≤ 0 corresponds to the Compensated law of demand for compensated price changes pt0 x(pt−1 , wt−1 ) = wt . The result above quantitatively connects violations
of this necessary condition for rationality with the error Slutsky matrix functions E π and E ν .
Next, we connect this to the Slutsky matrix norm.
Proposition 6. Consider any locally compensated demand path. For two demand functions x
and y such that their nearest rational Slutsky matrix function is the same S x,r (p, w) = S y,r (p, w),
it holds that for the weighting matrix W (t) =
||E x,π ||2W U
+
||E x,ν ||2W U
∂p(t) ∂x(t)
∂t
∂t
x,π
x,ν
≥
||E y,π ||2W U
+
∂p(t) ∂p(t) 0
∂t
∂t
||E y,ν ||2W U
and U the identity, one has that
if and only if
∂p(t) ∂x(t)
∂t
∂t
≥
∂p(t) ∂y(t)
∂t
∂t .
∂p(t) ∂y(t)
under the assumption S x,r (p, w) = S y,r (p, w) it holds that
∂t
∂t
∂p(t)
∂p(t)
∂p(t)
(t) + E (t)] ∂t ≥ ∂t [E y,π (t) + E y,ν (t)] ∂p(t)
∂t [E
∂t .
∂p(t) 0
∂p(t)
∂p(t) ∂p(t) 0
x,π
x,π
x,ν
Observe that T r( ∂t [E (t)+E (t)] ∂t ∂t [E x,π (t)+E x,ν (t)] ∂p(t)
(t)+
∂t ) = [ ∂t [E
∂p(t)
x,ν
2
E (t)] ∂t ] because the expression is a quadratic form, a scalar.
Proof. If
≥
By the cyclical properties of the trace:
0
0
∂p(t)
x,π
x,π
T r( ∂p(t)
(t) + E x,ν (t)] ∂p(t)
(t) + E x,ν (t)] ∂p(t)
∂t [E
∂t
∂t [E
∂t ) =
0
0
0
0
∂p(t)
∂p(t)
x,π
T r([E x,π (t) + E x,ν (t)] ∂p(t)
(t) + E x,ν (t)] ∂p(t)
∂t
∂t [E
∂t
∂t ).
And we recognize that:
∂p(t)
∂p(t)
x,π
x,π
(t) + E x,ν (t)] ∂p(t)
(t) + E x,ν (t)]||2M
T r([E x,π (t) + E x,ν (t)] ∂p(t)
∂t
∂t [E
∂t
∂t ) = ||W (t)[E
where W (t) =
∂p(t) ∂p(t) 0
∂t
∂t .
By an analogous argument to that in Theorem 1 above, ||W (t)[E x,π (t) + E x,ν (t)]||2M =
||W (t)[E x,π (t)]||2M + ||W (t)E x,ν (t)]||2M because W (t) =
for all t such that
∂p(t)
∂t
∂p(t) ∂p(t) 0
∂t
∂t
x,ν
6= 0 ∈ RL and W (t)E x,π (t), W (t)E
is a Positive Definite matrix
(t) are orthogonal.
This implies that ||E x,π ||2W U + ||E x,ν ||2W U ≥ ||E y,π ||2W U + ||E y,ν ||2W U .
Conversely, if ||E x,π ||2W U + ||E x,ν ||2W U ≥ ||E y,π ||2W U + ||E y,ν ||2W U and S x,r (p, w) = S y,r (p, w)
∂x(t) 2
∂p(t) ∂y(t) 2
it follows that [ ∂p(t)
∂t
∂t ] ≥ [ ∂t
∂t ] and this inequality is preserved under monotone
transformations so that
7
∂p(t) ∂x(t)
∂t
∂t
≥
∂p(t) ∂y(t)
∂t
∂t .
Examples and Applications
The rationality assumption has long been seen as an approximation of actual consumer behavior.
Nonetheless, to judge whether this approximation is reasonable, one should be able to compare
any alternative behavior with its best rational approximation. Our results may be helpful in
this regard, as the next examples illustrate.
7.1
The Sparse-Max Consumer Model of Gabaix (2014)
This model generates analytically tractable behavioral demand functions and Slutsky matrices.
In this example, we compare the matrix nearness distance to the “underlying rational” Slutsky
matrix function proposed by Gabaix and compare it to the one proposed here. This example
shows that there exists a rational demand function that is behaviorally closer to the sparse max
consumer demand proposed by Gabaix than the “underlying rational” model of his framework.
Consider a Cobb-Douglas model xCD (p, w) such that:
xCD
= αpiiw for i = 1, 2.
i
αi w
xCD
i,pi = − p2
i
20
xCD
i,w =
sCD
i,i =
sCD
i,j =
αi
pi
− αpi2w + αpii αpiiw
i
αi αj w
pi pj .
i )w
= − αi (1−α
p2
i
The Slutsky matrix
is:
" α function
#
α1 α2 w
1 α2 w
−
2
p1 p2
p1
CD
S
(p, w) =
.
α1 α2 w
− α1pα22 w
p1 p2
2
Let us denote Gabaix’s theory of behavior of the Sparse-max consumer by G. Then the
demand system under G is:
xG
i =
αi
w
P
pG
i
j αj
pj
pG
j
for i = 1, 2.
This demand system fulfills Walras’ law. This function has an additional parameter with
d
respect to xCD (p, w), the perceived price pG
i (m) = mi pi + (1 − mi )pi . The vector of attention
to price changes m, weighs the actual price pi and the default price pdi .
Consider
matrix of attention for the sparse-max consumer:
" the following
#
1 0
M=
0 0
That is, the consumer does not pay any attention to price changes in p2 , but perceives price
changes perfectly for p1 . One of Gabaix’s elegant results relates the Slutsky matrix function of
xG , to the Cobb-Douglas benchmark. The behavioral Slutsky matrix evaluated at default prices
(in all this example p = pd ) is:
CD
S G (p, w) = S
" (p, w)M #
− α1pα22 w 0
1
S G (p, w) =
.
α1 α2 w
0
p1 p2
This matrix is not symmetric, not NSD, nor singular with p in its null space. Applying
d
Theorem 1, the nearest
matrix when
# p = p is:
" Slutsky
α1 α2 w
α1 α2 w
− p2
p22
p1 p2
1
S r (p, w) = p2 +p
2
α1 α2 w
α1 α2 w
1
2
−
p1 p2
p22
Also, one has
E(p, w) = S(p, w) − S r (p, w)
#
"
[1 − b(p)][ α1pα22 w ] b(p)( αp11 αp22w )
1
E(p, w) = −
[1 − b(p)][ αp11 αp22w ] b(p)(− α1pα22 w )
2
with
b(p) =
p22
.
p21 +p22
Now, we compute a useful quantity:
T r(E 0 E) =
w2 α2 2 α21
.
p21 p22
It is convenient to compute the contributions of the violations of symmetry and singularity
in p separately.
0
0
T r(E 0 E) = T r(E σ E σ ) + T r(E π E π ) =
2
2 2
1 w α2 α1
2 p21 p22
+
2
2 2
1 w α2 α1
2 p21 p22 .
In this case, regardless of
the values that w takes, the contribution of each kind of violation is equal and amounts to
´w
exactly half of the total distance. In fact, we have: ||E||2 = 12 w T r(E σ (w)0 E σ (w))dw +
´
α2 2 α21
w3 −w3
1 w
π
0 π
, with p = pd .
2 w T r(E (w) E (w))dw =
3
p2 p2
1 2
Note, however, that in this example the third component of the violations, the one stemming
from NSD, is zero when the prices are evaluated at the default. Since the behavioral model
proposed by Gabaix does not satisfy WARP and its Slutsky matrix function violates NSD, we
conclude that the violation of the WARP is not massive, in fact, it affects the size of the Slutsky
21
δ(pd2 , α)
1.0
0.8
0.16
0.12
0.6
0.08
0.4
0.04
0.2
-6
-4
-2
0246
0.0
1.0
1.2
1.4
1.6
1.8
2.0
Figure 2: Level curves sparse-max consumer example . The figure plots α ∈ [0, 1] on its vertical
axis and pd2 ∈ [1, 2] on its horizontal axis.
norm only through its interactions with homogeneity of degree zero or “money illusion”. Our
approach can also be used in the general case. We compute the previous quantities at any p,
and any pd with m = [1, 0]0 .
T r(E 0 E) =
2
[pd
w2 α2 2 α21
2]
4,
p21
[p2 +[pd
−p
2 ]α1 ]
2
0
0
with T r(E σ E σ ) = T r(E π E π ) and E ν = 0.
The expression above has a positive derivative with respect to pd2 for α1 +α2 = 1, this indicates
that
∂
δ(·)
∂pd
2
> 0 for any p. A sparse consumer that does not pay attention to p2 will be further
from rationality when the default price pd2 is high. Furthermore, the power of our approach lies
in the decomposition of ||E||2 = ||E σ ||2 + ||E π ||2 . In this case, the decomposition suggests that
the violation of WARP can be seen as a byproduct of the violations of symmetry and singularity
stemming from the “lack of attention” to price changes of good 2 and the “nominal illusion” or
lack of homogeneity of degree zero in prices and wealth in such a demand system.15
To show the tractability of our approach in this example, we will study a very simple region Z, with the aim of illustrating how one can learn from the effect of a behavioral pa1
d
d
rameter such
as α1 and p2 . Let Z = {w, p1 = 1, p2 ∈ [1, 2]}, then δ(α1 , p2 ) = 3 (α1 −
1
1
1)α1 2 [pd2 ]2 (α1 (pd −2)+2)
. One can now visualize this δ in the α, pd2 space,
3 − (α (pd −1)+1)3
1
2
2
that is at pd2 ∈ [1, 2] and α1 ∈ [0, 1] (figure 2). We can observe that α1 has a non-linear effect on
δ, and the distance toward the rational matrix goes to zero when either α1 → 0 or α1 → 1 for
all pd2 ∈ [1, 2].
To finish this example we study the errors in comparative statics analysis. This is done
locally.16 Fix (p, w) = (pd , w) as the reference point. First, we compare the distance between S G
15 This
observation is robust to other specifications (e.g. CES family).
general, our approach does not require finding the closest xr ∈ R(Z) to xG , nevertheless doing so may
help to complement the understanding of how much this particular bounded rationality model differs from the
16 In
22
and S CD , in terms of errors in comparative statics analysis. If we use xCD as an approximation
of xG at the default prices, we obtain the following local comparative statics error ||S G (pd , w) −
S CD (pd , w)||2M =
2
d 2
(pd
w2 α2 2 α21
1 ) +(p2 )
d 2.
2
2
(pd
(pd
2)
1 ) (p2 )
By our results in Subsection 6.1, we can decompose
this quantity as the sum of a mis-specification error and ||E(pd , w)||2M . In fact, ||S CD (pd , w) −
S G (pd , w)||2M = ||S CD (pd , w) − S r (pd , w)||2M + ||E(pd , w)||2M , with ||S CD (pd , w) − S r (pd , w)||2M =
w2 α2 2 α21
4
(pd
2)
w2 α2 2 α21
d 2.
2
(pd
1 ) (p2 )
and ||E(pd , w)||2M =
We know also, by the results in Subsection 6.1, that even if we could improve the misspecification error, the Slutsky matrix error norm ||E(pd , w)||2M will not change at all. To illustrate this, we find the Cobb-Douglas demand that minimizes the error in comparative statics
analysis at the reference price-wealth pair (pd , w). We can write this problem parametrically,
through characterizing the Cobb-Douglas family by a parameter β ∈ [0, 1]: xCD2 (p, w, β) =
(1−β)w 0
) . We solve the problem β ∗ ∈ argminβ∈[0,1] ||S CD2 (pd , w, β) − S G (pd , w)||2M . There
( βw
p1 ,
p2
√ d0 d d 2 d 2
p p [(p1 ) +(p2 ) [1−2α1 ]2 ]
are two solutions β ∗ = 12 ±
, which depend on the given parameters
2pd0 pd
and the fixed default price. The mis-specification error is zero, that is ||S CD2 (pd , w, β ∗ ) −
S r (pd , w)||2M = 0. Because of this, in this case, the error in comparative statics reduces to the
behavioral error captured by the Slutsky norm: ||S CD2 (pd , w, β) − S G (pd , w)||2M = ||E(pd , w)||2M ,
where ||E(pd , w)||2M =
w2 α21 α22
d 2.
2
(pd
1 ) (p2 )
Observe that it does not depend on β ∗ and remains unchanged with respect to the case of
S CD . Of course, even if we used the “best” rational model to minimize errors in comparative
statics analysis locally, we would not reduce at all the Slutsky matrix error norm that stems
from lack of rationality as captured by properties σ, π, ν.
7.2
Hyperbolic Discounting
The literature on self-control and hyperbolic discounting has flourished in macroeconomics and
development economics. In this example, we study a three-period model that allows us to
illustrate the use of our methodology. Our aim is to measure the violations of property R by
naive and sophisticated quasi-hyperbolic discounters.
The optimization problem for a consumer that can pre-commit is: max{xpi }i=1,2,3. u(xp1 ) +
P3
βθu(xp2 ) + βθ2 u(xp3 ) subject to the budget constraint i=1 pi xpi = w.
1
1
Which gives the demand system: (i) xp1 = [p1 + p2 [βθ pp21 ] σ + p3 [βθ2 pp31 ] σ ]−1 w. (ii) xp2 =
1
1
[βθ pp21 ] σ xp1 (iii) xp3 = [βθ2 pp13 ] σ xp1 .
The naive quasi-hyperbolic discounter will have the following demand system:
In the first period, the consumer assumes he will stick to his commitment in the second
period and consumes the same amount as in the pre-commitment case (i.e. xh1 = xp1 ). However,
when period two arrives, he re-optimizes taking as given the remaining wealth w − p1 xh1 : xh2 =
1
w−p1 xh
1
and xh3 = [βθ pp23 ] σ xh2 .
p2 1
p2 +p3 [βθ p ] σ
3
The analytical result for the matrix nearness norm has a nice expression:
(σ − 1)2 w2 p21 + p22 + p23
T r(E 0 E) =
2σ 2
βθ2 p1 σ1
p3
−
βθp1
p2
σ1 βθp2
p3
σ1 2
2 4 ,
1
1
2 σ1
βθp2 σ
βθp1 σ
βθ p1
p3 p3
+ p2
p 2 p2
+ p3
+ p1
p3
standard rational one in practical terms
23
δ(β, θ)
1.0
0.8
0.12
0.10
0.6
0.08
0.06
0.04
0.4
0.02
-6
-4
-2
0246
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Figure 3: Level curves hyperbolic discounter example . The figure plots β ∈ [0, 1] on its horizontal
axis and θ ∈ [0, 1] on its vertical axis.
which readily gives us that: (i) when σ = 1 then T r(E 0 E) = 0 and δ = 0, that is the demand
is rational, (ii) when β = 1 then T r(E 0 E) = 0, (iii) finally when β → 0, θ → 0, then δ → 0.
In these three cases by the previous results → 0. In fact, in the limit cases the hyperbolic
demand system is rational. Take for instance case (iii), because the agent consumes everything
in the first period and gives no weight to the other time periods then it is trivially rational, with
S r → 0 and xh1 →
w
p1
and xh2 , xh3 → 0. In case (i), the logarithmic utility case, the hyperbolic
discounter manages to keep his commitment and therefore his consumption is time consistent
and ||E|| = 0.
To illustrate further the use of the tools developed here, we find an explicit value for δ
in terms of the behavioral parameters β, θ, for an arbitrary rectangle Z of prices and wealth.
Consider the region Z = {p1 , p2 , p3 = 1, w ∈ [1, 2]} , with σ =
1
2
we compute δ(β, θ), which
can be represented graphically in the box β ∈ [0, 1], θ ∈ [0, 1]. The analytical expression for
2
7β 4 (β 2 −1) θ 8
δ 2 = 2(β 2 θ2 +1)2 (β 2 (θ4 +θ2 )+1)4 .
The level curves (figure 3) show that the hyperbolic discounter is very close to the rational
consumer, in the matrix nearness sense, for very low values of β, θ and for values of θ ≤ 21 .17 This
makes intuitive sense as a lower θ means heavier discount on the future and lower consumption
of goods of time 2 and 3 that are the ones affected by self-control.
Another observation that we can draw from this example is that for any arbitrary compact
region Z of prices and wealth analyzed ||E||2 = ||E σ ||2 . That is, only the asymmetric part
plays a role in the violation of property R. In other words, under this numerical conditions
17 One can also use the level curves in this example to identify pairs (β, θ) that are “equidistant” from rationality,
capturing an interesting tradeoff between the short-run and the long-run discount factors and its effects on the
violations of the Slutsky conditions.
24
the hyperbolic discounter violates symmetry but it satisfies singularity in prices and negative
semidefiniteness. The hyperbolic discounter fulfills WARP.
7.3
Sophisticated Quasi-Hyperbolic Discounting
The sophisticated quasi-hyperbolic discounter is intuitively closer to rationality. However, the
Slutsky norm helps appreciate some of the subtleties and assess which conditions of rationality
are fulfilled by this type of consumer. We build this example as a followup to the naive quasihyperbolic consumer. In this case, the consumer knows that in t = 2 he will not be able to
keep his commitment and therefore will adjust his consumption at t = 1. Then the consumer
maximizes
h
2
h
maxxsh
u(xsh
1 ) + βθu(x2 ) + βθ u(x3 )
1
where xh2 , xh3 are known to his in t = 1 and depend on period 1 consumption. However, he
can control only how much he consumes in the first period. Then, the first period consumption
under sophisticated hyperbolic discounting is:
xsh
1

−1
 σ1
1−σ
 p1 βθ + p1 βθ2 [βθ pp23 ] σ 

=
i1−σ  + p1 
 h
 w
1
p2 + p3 [βθ pp23 ] σ
The argument in the integral of the expression for δ for a generic Z is given by the quantity:
T r(E 0 E) =
2
(β − 1)2 (σ − 1)2 w2 θ4/σ p21 + p22 + p3

σ1 σ2 −2
2

βp2 + p3 βθp
p3
2/σ

βp1 1
βθp
p3 p3 p 2 σ +p2
3
4
 1
σ−1 1
1 σ
βθp
βθp
θp1 p3 p 2 σ +p2
βp2 +p3 p 2 σ
3
3

 + p1 
2σ 4 

p2
As expected, this implies that: (i) when σ = 1, δ = 0 for any Z; (ii) when β = 1, δ = 0;
and (iii) when β = 0, δ = 0. Thus, in all these cases, = 0. Also, the decomposition of
||E||2 = ||E σ ||2 , which means that only the symmetry property is violated, while the weak
axiom and the homogeneity of degree zero in prices and wealth are preserved.
Finally, we compare this quantity with the case of the naive hyperbolic discounter. We
consider the ratio r =
T r(E sh0 E sh )
.
T r(E h0 E h )
When r < 1, the sophisticated hyperbolic consumer has
a lower δ for any Z and any parameter configuration. The first finding is that the ratio r
and the implied values of δ depend crucially on the parameter σ. For σ = 1, both E sh , E h
are equal to zero: this is a knife-edge case, in which the marginal rates of substitution yield
optimal consumptions equal to the commitment baseline. For σ < 1 (e.g., 1/2), the sophisticated
hyperbolic discounter has a uniformly lower δ and thus r < 1.
However, for σ > 1 (e.g., equal to 2), the naive hyperbolic discounter has a uniformly
25
lower δ, hence r > 1. Although this may seem counterintuitive, it tells us that the closest
rational type (which need not be the commitment baseline) is closer for the naive than it is for
the sophisticated consumer. To see this, note that the “rational” comparative statics analysis
fails due to two possible sources: (i) The lack of self-control in the consumption of period 2
(overspending); and (ii) in the case of the sophisticated type, an additional effect due to the
strategic change in period 1 consumption (overspending or underspending). So, the “rational”
comparative statics analysis gives a higher failure than in the naive case for σ > 1, because for
an increase in prices for the naive case the change in period 1 consumption equals that in the
rational case and only the second and third period demanded amounts are different. In contrast,
for the sophisticated case, demanded amounts for the three goods all differ, in particular due
to a higher level of wealth remaining at the end of period 1, leading to a higher overspending
in period 2. This observation holds for such CRRA instantaneous utility with any parameter
values of δ,θ ∈ [0, 1].18
8
Literature Review
The canonical treatment of measuring deviations from rational consumer behavior was establish
by Afriat (1973) with its critical cost-efficiency index. Afriat’s index measures the amount by
which budget constraints have to be adjusted so as to eliminate violations of the Generalized
Axiom of Revealed Preference (GARP). Varian (1985; 1990) refines Afriat’s measure by focusing
on the minimum adjustment of the budget constraint needed to eliminate violations of GARP.
Houtman and Maks (1985) measure deviations from GARP through identifying the largest subset
of choices that is consistent with maximizing behavior. More recently, Echenique, Lee and Shum
(2011), give a new measure of violations of revealed preference behavior called the “money pump
index” . Also Jerison and Jerison (2012) propose a way to bound Afriat’s index of cost-efficiency
using an index of violations of the symmetry and negative semidefiniteness Slutsky conditions.
It would be interesting to compare our Slutsky matrix norm with these other approaches.
The closest treatment of the problem to our work is the approximately rational consumer
demand proposed by Jerison and Jerison (1992; 1993). These authors are able to relate the violations of negative semidefiniteness and symmetry of the Slutsky matrix to the smallest distance
between an observe smooth demand system and a rational demand. Russell (1997) proposes a
notion of quasi-rationality. Russell’s argument links the Slutsky matrix anti-symmetry part with
the lack of integrability of a demand system.
Our work takes a different methodological approach to this problem and generalizes the
results to the case of violations of singularity of the Slutsky matrix. More importantly, this new
approach allows to treat the three kinds of violations of the Slutsky conditions simultaneously.
For instance, new behavioral models like the sparse-max consumer Gabaix (2014) suggest the
presence of a money illusion such that prices are not in the null space of the Slutsky matrix. We
also emphasize a positive measure of bounded rationality based on errors in making comparative
18 Furthermore, to enhance the comparison for the case of σ = 1 , we compute explicitly the expression for
2
δ in the same region Z = {p1 , p2 , p3 = 1, w ∈ [1, 2]} as in the previous example for the naive discounter:
2
δ 2 (β, θ) = [14(β − 1)2 β 6 θ8 βθ2 + 1 ]/[β 2 θ2 βθ2 βθ2 + 2 + 2 + 1]4 . The level curves of this δ expression are
very similar to the naive case, but it is closer to rationality uniformly for all parameter configurations. Evaluated
at the values of β = 0.7 and θ = 0.9, one gets the value δ = 0.0703847, which is slightly lower than the δ of
the naive hyperbolic case for the same σ = 21 (δ = 0.074), where the parameters are taken from the empirical
literature.
26
statics analysis when one assumes (perhaps incorrectly) that the consumer is rational. The
only other work we are aware of that analyzes comparative statics with behavioral consumers is
Farhi & Gabaix (2015). In their interesting work they propose alternate ideas to the traditional
Slutsky matrix including extensions of the Slutsky matrix for nonlinear budget constraints. It
remains an open task how to extend our methodology to these new concepts.
9
Conclusion
By redefining the problem of finding the closest rational demand to an arbitrary observed behavior in terms of matrix nearness, we are able to pose the problem in a convex optimization
framework that permits a better computational implementability and provide a tractable approach with a closed form solution. We define a metric in the space of smooth demand functions
and finally propose a way to recover the best Slutsky approximation matrix function under a
Frobenius norm. Our approach gives a geometric interpretation in terms of transformations of
the Slutsky matrix or first order behavior of demand functions. As a result, a classification of
the different kinds of violations of rationality is also provided. Comparative statics for a boundedly rational consumer is measured with our norm. Our approach is also suited to measure the
mis-specification error from assuming a given form of rationality when the consumer’s behavior
is actually not rational.
References
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28
Appendix: Proofs
We begin this appendix with the following claim, which is an auxiliary result to be used in the
sequel.
Proof of Claim 1
Proof. The problem is minS r ||S − S r || subject to S r (z) ≤ 0, S r (z) = S r (z)0 , S r (z)p = 0 for
z ∈ Z.
Under the Frobenius norm, the minimization problem amounts to finding the solution to
´
minS r z∈Z T r([S(z) − S r (z)]0 [S(z) − S r (z)])dz
subject to the stated constraints.
The objective function is strictly convex, because of the use of the Frobenius norm. This
norm is also a continuous functional.
The constraint set MR (Z) is convex and closed. In fact, the cone of negative semidefinite
matrices is a closed and convex set. Also, the set of symmetric matrices is closed and convex,
and finally the set of matrices with eigenvalue λ = 0 associated with eigenvector p is convex.
To see the last statement, let A(z)p = 0, B(z)p = 0, and let C(z) = αA(z) + (1 − α)B(z) for
α ∈ (0, 1). It follows that C(z)p = 0. Then MR (Z) is the intersection of three convex sets and
is therefore convex itself. It is also useful to note that all three constraint sets are subspaces of
M(Z) and the intersection MR (Z) is itself a subspace of M(Z).
Now we prove that not only the symmetric and the NSD constraints sets are closed but so
is the set of all MR (Z). Any matrix function in the constraint set is a symmetric NSD matrix
with p in its null space. Therefore, every sequence of matrix functions in the constraint set
has the form Dn (z) = Qn (z)Λn (z)Qn (z)0 , where Λn (z) = Diag[λni (z)]i∈1,...L with ascending
ordered eigenvalues functions. It follows that the eigenvalue function in position L, L is the
null eigenvalue λL = 0, or the null scalar function. That is, imposing an increasing order
the position 1, 1 is then held by λn1 (z) ≤ λn2 (z) ≤ . . . ≤ 0 where the order is induced by the
distance to the null function using the Euclidean distance for scalar functions defined over Z.19
The matrix function Qn (z) = [q1n · · · p] is the orthogonal matrix with eigenvectors functions
n
as columns. For all Dn (z) ∈ MR (Z), λnL = 0 is associated with the price vector qL
= p
always, to guarantee that p is in its null space. The eigenvectors are defined implicitly by the
condition Dn (z)qin (z) = λn (z)qin (z) with pointwise matrix and vector multiplication and qin ⊥ p
or hqin , pi = 0 using the inner product for C 0 (Z) for i = 1, . . . , L − 1 and for all n ∈ N. Take any
sequence of {Dn (z)}n∈N with Dn (z) ∈ MR (Z) for each z ∈ Z, with limit limn→∞ Dn (z) = D(z).
We want to show that D(z) ∈ MR (Z). It should be clear that any Dn (z) → D(z) converges to
a symmetric matrix function (the symmetric matrix subspace is an orthogonal complement of a
subspace of M(Z) (the subspace of skew symmetric matrix functions) and therefore, it is always
closed in any metric space). It is also clear that D(z)p = 0 since λnL = 0 for all n and certainly
n
n
λnL → 0 with the associated eigenvector qL
= p for all n and qL
→ p. This condition, along with
symmetry, guarantees that qi6n=L → q ⊥ p. Finally, the set of negative scalar functions is closed.
Then, λni6=L → λ(z)− with λi6=L (z)− = min(0, λi6=L (z)). This is a negative scalar function by
construction, since if λi6=L (z) > 0 then λi6=L (z)− = 0. Then MR (Z) is closed. Observe that
19 These eigenvalue functions can be labeled because the domain Z is simply connected. The order admits
crossing eigenvalue functions.
29
MR (Z) is closed since it is in the intersection of three closed sets. In conclusion, since the
Frobenius norm in M(Z) is a continuous and strictly convex functional and the constraint set
is closed and convex, the minimum is attained and it is unique.
Next, we present a claim that is auxiliary to establish Lemma 2, in the proof of Theorem 1.
Claim 3
Claim 3. The matrix E π (z) is pointwise orthogonal to S(z)+ . That is T r(E π (z)0 S(z)+ ) = 0.
Proof. By definition S σ,π (z) = S σ (z) + E π (z), with E π (z) a symmetric matrix such that
E π (z)p 6= 0 when S σ (p)p 6= 0 and E π (z) = 0 when S σ (p)p = 0. Thus, E π (z) is always
singular.
One can then write the direct sum decomposition of the set A(z) of symmetric singular
matrix functions with the property that p0 A(z)p = 0 as follows: A(z) = P(z) ⊕ N (z) for all
z ∈ Z, where
P(z) = {E π (z) : T r(E π (z)pp0 ) = 0
and
E π (z)p 6= 0
for E π (z) 6= 0}
and
N (z) = {N (z) : T r(N (z)pp0 ) = 0,
N (z)p = 0}.
To see that this is a direct sum decomposition, first observe that P(z) ∩ N (z) = {0}, with
0 denoting the zero matrix function, by construction. Furthermore, any A(z) ∈ A(z) can be
written as a sum of A(z) = E π (z) + N (z) since A(z)p = 0 or (exclusive) A(z)p 6= 0, for A(z) 6= 0.
Furthermore, p0 A(z)p = p0 E π (z)p + p0 N (z)p = 0 for any E π (z), N (z). Then the decomposition
is precisely A(z) = E π (z) when A(z)p 6= 0 and A(z) = N (z) when A(z)p = 0. Since every
direct sum decomposition represents the sum of a subspace and its orthogonal complement,
andN (z) is a subspace in the space of symmetric matrix-valued functions, it follows that P(z)
is its orthogonal complement. In particular, since S(z)+ p = 0 and T r(S(z)+ pp0 ) = 0, it follows
that T r(E π (z)S(z)+ ) = 0, for z ∈ Z. f
30